derivative of 10^6x^2e^{-x}

解答

\frac{d}{d x} (1 0^{6} x^{2} e^{- x})

1000000 (- e^{- x} x^{2} + 2 e^{- x} x)

求解步骤

\frac{d}{d x} (1 0^{6} x^{2} e^{- x})

化简

1 0^{6} x^{2} e^{- x} : 1000000 e^{- x} x^{2}

= \frac{d}{d x} (1000000 e^{- x} x^{2})

将常数提出:

(a \cdot f)^{'} = a \cdot f^{'}

= 1000000 \frac{d}{d x} (e^{- x} x^{2})

使用微分乘法定则:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = e^{- x}, g = x^{2}

= 1000000 (\frac{d}{d x} (e^{- x}) x^{2} + \frac{d}{d x} (x^{2}) e^{- x})

\frac{d}{d x} (e^{- x}) = - e^{- x}

\frac{d}{d x} (x^{2}) = 2 x

= 1000000 ((- e^{- x}) x^{2} + 2 x e^{- x})

化简

= 1000000 (- e^{- x} x^{2} + 2 e^{- x} x)

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